When I add a cylinder, it doesn't even show up on my screen at all
The mesh will be added in the position of the 3D cursor. You need to place the 3D cursor where you want the cylinder to appear.
In the meantime you have got the mesh somewhere - it showed up in the Outliner when you pressed Add cylinder.  You can find the object while it's still selected by pressing Numpad . (dot). The viewport camera will zoom in to the selected object. Or, if it's not selected, select it in the Outliner.